Numerical approach to the Schrödinger equation in momentum space

Karr, William A.; Jamell, C. R.; Joglekar, Yogesh N.

doi:10.1119/1.3272021

Cited by 7 publications

(11 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…They are quite accurate but resort finally to numerical integrations on a mesh. Direct computations on a mesh are easier to carry out, but they require very large mesh if a good quadrature rule is not used [14]. As we will see, the LMM in momentum space is very easy to implement and can also give accurate results.…”

Section: Introductionmentioning

confidence: 94%

“…But, for some particular problems, it can be preferable to work in the momentum space. This is the case when the potential presents discontinuities in the configuration space [14] or when the potential is given in the momentum space. In this last case, if it is possible to use the LMM by computing first the Fourier transform of the potential, we will show here that the LMM can be adapted to solve the eigenequations directly in momentum space.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Lagrange-mesh calculations in momentum space

2012

View full text Add to dashboard Cite

The Lagrange-mesh method is a powerful method to solve eigenequations written in configuration space. It is very easy to implement and very accurate. Using a Gauss quadrature rule, the method requires only the evaluation of the potential at some mesh points. The eigenfunctions are expanded in terms of regularized Lagrange functions which vanish at all mesh points except one. It is shown that this method can be adapted to solve eigenequations written in momentum space, keeping the convenience and the accuracy of the original technique. In particular, the kinetic operator is a diagonal matrix. Observables in both configuration space and momentum space can also be easily computed with a good accuracy using only eigenfunctions computed in the momentum space. The method is tested with Gaussian and Yukawa potentials, requiring respectively a small or a great mesh to reach convergence.

show abstract

Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Lagrange-mesh calculations in momentum space

2012

View full text Add to dashboard Cite

show abstract

“…For most potential function the equation has to be solved by different suitable ways. Several lines of approach have been followed in the study of SE for different types of potentials such as: variational [1,2] and perturbational schemes [3], also combined with direct numerical methods [4] and series solutions [5] as well as the recently approach to solve SE into momentum representation [6]. Also there is a geometic approach to solve SE, known as geometric quantisation [7].…”

Section: Introductionmentioning

confidence: 99%

A Numerical Approach for the Solution of Schrödinger Equation With Pseudo-Gaussian Potentials

Iacob

Lute

Iacob

2015

Annals of West University of Timisoara - Physics

View full text Add to dashboard Cite

The Schrödinger equation with pseudo-Gaussian potential is investigated. The pseudo-Gaussian potential can be written as an infinite power series. Technically, by an ansatz to the wave-functions, exact solutions can be found by analytic approach [12]. However, to calculate the solutions for each state, a condition that will stop the series has to be introduced. In this way the calculated energy values may suffer modifications by imposing the convergence of series. Our presentation, based on numerical methods, is to compare the results with those obtained in the analytic case and to determine if the results are stable under different stopping conditions.

show abstract

“…These ensembles arise from a fundamental constraint: the Hermiticity of the random Hamiltonian, that guarantees real eigenvalues, and eigenvectors that are orthogonal with respect to the standard inner-product in quan-tum theory [4]. However, a large class of non-Hermitian matrices -parity and time-reversal symmetric Hamiltonians [5][6][7], rate-equation matrices [8], central potentials in momentum space [9], etc. -has real spectra although the eigenvectors are not orthogonal under the standard inner-product [5].…”

Section: Introductionmentioning

confidence: 99%

“…Such a disorder can result from sputtering deposition (for electrons) or from a circular patterned grating (for light). The resulting momentum-space Hamiltonian for such a potential is not Hermitian; but it has purely real eigenvalues [9]. Motivated by the generalization of this example to D-dimensions, in this paper, we only consider a diagonal inner product F :…”

Section: Introductionmentioning

confidence: 99%

Level density and level-spacing distributions of random, self-adjoint, non-Hermitian matrices

Joglekar

Karr

2011

Phys. Rev. E

Self Cite

View full text Add to dashboard Cite

We investigate the level density σ(x) and the level-spacing distribution p(s) of random matrices M = AF ≠ M{†}, where F is a (diagonal) inner product and A is a random, real, symmetric or complex, Hermitian matrix with independent entries drawn from a probability distribution q(x) with zero mean and finite higher moments. Although not Hermitian, the matrix M is self-adjoint with respect to F and thus has purely real eigenvalues. We find that the level density σ{F}(x) is independent of the underlying distribution q(x) and solely characterized by F, and therefore generalizes the Wigner semicircle distribution σ{W}(x). We find that the level-spacing distributions p(s) are independent of q(x), and are dependent upon both the inner product F and whether A is real or complex, and therefore generalize the Wigner surmise for level spacing. Our results suggest F-dependent generalizations of the well-known Gaussian Orthogonal Ensemble and Gaussian Unitary Ensemble classes.

show abstract

Numerical approach to the Schrödinger equation in momentum space

Cited by 7 publications

References 15 publications

Lagrange-mesh calculations in momentum space

Lagrange-mesh calculations in momentum space

A Numerical Approach for the Solution of Schrödinger Equation With Pseudo-Gaussian Potentials

Level density and level-spacing distributions of random, self-adjoint, non-Hermitian matrices

Contact Info

Product

Resources

About